Title: Positivity of the Limit F-signature and the Frobenius-Alpha Invariant

Abstract: Tian introduced the alpha-invariant in 1987 to detect the existence of Kähler–Einstein metrics on Fano varieties. It has since played a central role in the study of Fano varieties and K-stability. In this talk, we will discuss a positive characteristic analog of the alpha-invariant, which is defined via the properties of the Frobenius map and the theory of F-singularities. In positive characteristics, the Frobenius-Alpha invariant is also closely related to another asymptotic invariant of singularities called the F-signature.
The main results concern the behavior of the Frobenius-alpha invariant and the F-signature under the process of reduction to characteristic p >> 0 of a fixed complex Fano variety. Motivated by applications to the sizes of local fundamental groups, Carvajal-Rojas, Schwede and Tucker conjectured that the F-signatures remain uniformly bounded away from zero when we reduce a complex KLT singularity to large characteristics. We will present joint works with Yuchen Liu, and with Anna Brosowsky, Izzet Coskun and Kevin Tucker, in which we prove this conjecture in many new cases including for cones over low degree smooth hypersurfaces and most three dimensional KLT singularities. We will present some of the key ideas in the proof, which come from recent results in the K-stability theory of Fano varieties.